Maximal function on generalized Lebesgue spaces $L^{p(\cdot)}$

TLDR: In this paper, the Hardy-Littlewood maximal function on the generalized Lebesgue space Lp(·)(Rd) under a continuity assumption on p that is weaker than uniform Holder continuity was shown to be bounded.

Abstract: We prove the boundedness of the Hardy–Littlewood maximal function on the generalized Lebesgue space Lp(·)(Rd) under a continuity assumption on p that is weaker than uniform Holder continuity. We deduce continuity of mollifying sequences and density of C∞(Ω) in W1,p(·)(Ω) . Mathematics subject classification (2000): 42B25, 46E30.

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M athematical
I nequalities
& A pplications
Volume 7, Number 2 (2004), 245–253
MAXIMAL FUNCTION ON GENERALIZED LEBESGUE SPACES L
p(·)
L. DIENING
Abstract. We prove the boundedness of the Hardy–Littlewood maximal function on the general-
ized Lebesgue space L
p(·)
(R
d
) under a continuity assumption on p that is weaker than uniform
H
¨
older continuity. We deduce continuity of mollifying sequences and density of C
∞
(
Ω)
in W
1,p(·)
(Ω) .
Mathematics subject classification (2000): 42B25, 46E30.
Key words and phrases: maximal function, generalized Lebesgue spaces, generalized Orlicz spaces,
mollifier, electrorheological fluids.
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